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x
f (u) = eu
f (u) = 1
u
f (u) = au
f (u) = sen(u)
f (u) = cos(u)
f (u) = tg(u)
f (u) = cotg(u)
f (u) = sec(u)
f (u) = cosec(u)
f (u) = sec2(u)
f (u) = cosec2(u)
f (u) = sec(u)tg(u)
f (u) = cosec(u)cotg(u)
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f (x) ddx
[f (x)] = f ′(x) d(f (x))
= f ′(x)dx
d (f (x)) = f ′(x)dx
f (x)
f (x)
f (x)
∫
f ′(x)dx = f (x)
+ C
C
f ′(x)dx
f (x) C
f (x)
f ′(x) f (x)
C
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f (x) = x3 f ′(x) = 3x2
3x2dx = x3 + C
F (x)
f (x)
I
x ∈ I
F ′(x) = f (x)
F (x) = x3
3 f (x) = x2
F (x) =
x3
3
F ′(x) =
3x2
3
= x2 = f (x).
f (x)dx =
x2dx =
x3
3
d
dx (F (x)) =
d
dx
x3
3
=
3x2
3 = x2.
F (x) f (x)
F (x) + C
f (x)
f (x)
f (x)dx = F (x) + C,
x
f (x)
∫
F (x)
C
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f (x) = 2x
F (x) = x2, F ′(x) =
2x = f (x).
F (x) = x2 + 1, F ′(x) =
2x = f (x).
F (x) = x
2
− 1,
F ′(x) = 2x = f (x).
y
f (x) = cos(x)
d
dx [sen(x)] = cos(x) sen(x)
cos(x).
d
dx [sen(x) + 3] = cos(x) sen(x) + 3
cos(x).
d
dx [sen(x) − 8] = cos(x) sen(x) − 8
cos(x).
d
dx [sen(x) + k] = cos(x) sen(x) +
k cos(x)
k
cos(x)
sen(x) + k
1
3x3
1
3x3
+ 2
1
3 x3
− 5 1
3 x3
+ √ 2 f (x) = x2
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d
dx
1
3x3
= x2
d
dx
1
3x3 + 2
= x2
d
dx1
3 x3 − 5
= x2
d
dx
1
3x3 +
√ 2
= x2
f (x)
F (x) + C
f (x)
f (x) = y
ydx = yx + C
f (x) = y
x
y
f g I → R
K
K f (x)
K Kf (x)dx =
K
f (x)dx.
[f (x) ± g(x)] dx =
f (x)dx ± g(x)dx.
d
f (x)dx
= f (x)dx.
d
dx
f (x)dx
= f (x).
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X
x
x x
xndx =
xn+1
n + 1 + C, n ̸= −1.
xn
d
dx
xn+1
n + 1 + C
= (n + 1)
xn+1−1
n + 1 = xn.
dx = x + C
f (x) = 1 f (x) = x0
x0dx =
x0+1
0 + 1 + C = x + C
I a =
(x3 + 5x7)dx
I b = 1x3
+ 6
x√ x + 3 dx
I c =
3√
x2 + 4√
x√ x
dx.
I c ii)
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I c =
3√
x2√ x
dx +
4√
x√ x
dx ii
= x2
3
x1
2
dx + x1
4
x1
2
dx
=
x(
2
3− 1
2)dx +
x(
1
4− 1
2)dx
=
x
1
6 dx +
x−
1
4 dx
= x
1
6+1
16
+ 1 +
x−1
4+1
−14
+ 1 + C
x
I c = 6x
7
6
7 +
4x3
4
3 + C.
F f I
g
F g F ◦ g
= F (g(x))
[F (g(x))]′ = F ′(g(x))g′(x)
= f (g(x))g′(x).
F (g(x))
f (g(x))g′(x)
f (g(x))g′(x)dx = F (g(x)) +
C,
u = g(x)
du = g ′(x)dx
f (g(x))g′(x)dx =
f (u)du = F (u) + C.
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u = g(x)
du = g ′(x)dx
undu =
un+1
n + 1 + C, n ̸= −1
x x u
u x du
u = g(x) du = g′(x)dx
[g(x)]ng′(x)dx =
[g(x)]n+1
n + 1 + C, n ̸= −1
undu =
un+1
n + 1 + C, n ̸= −1.
I 1 =
(9x2 − 1) 13 xdx
) I a =
3√
2x + 1 dx I b =
cos(3x)
√ sen(3x) + 2dx I c =
sen(2x)
√ 1 + sen2(x)dx
I d =
ln2(x)
x dx I e =
3x
(a + bx2)3dx I f =
sen2(x) cos(x)dx
I g =
sec(3x)
1 + tg(3x)
2dx
I h =
arcsen(x)√
1 − x2 dx I i =
2 + ln(x)
x dx
I c =
sen(2x)
√ 1 + sen2(x)dx
I c
u = 1 + sen2(x).
du = 2sen(x)cos(x)dx = sen(2x)dx.
I c u
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I n =
du√
udu =
u−
1
2 du
= u−
1
2+1
−12
+ 1 + C
= 2u1
2 + C
= 2√
u + C.
u = 1 + sen2(x)
x u I n
I c
I c = 2 1 + sen2(x) + C.
I e =
3x
(a + bx2)3dx,a ̸= 0, b ̸= 0
I e
u = a + bx2.
du = 2bxdx
xdx =
du
2b
b ̸= 0.
I e u
I n =
3u3
du
2b = 3
2b
u−3du
i
=
3
2b
u−3+1
−3 + 1 + C
=
3
2b
u−2
−2 + C
= − 34bu2
+ C.
u = a + bx2
x u I n
I e
I e = − 34b(a + bx2)2
+ C.
I f =
sen2(x)cos(x)dx
I f
u = sen(x).
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du = cos(x)dx. I f
u
I n =
u2du =
u2+1
2 + 1 + C
= u3
3 + C.
u = sen(x) x
u I n I f
I f = sen3(x)
3 + C.
I h = 2 + ln(x)x
dx
I h
u = 2 + ln(x).
du = 1
xdx. I h
u
I n =
udu =
u1+1
1 + 1 + C
= u2
2 + C.
u = 2+ln(x) x
u I n I h
I h = (2 + ln(x))2
2 + C.
u
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) I a =
√ x − 1√
x
dx
) I b =
(5x + 2)4dx
) I c =
√ x2 + x4
dx
I d = x2(x2 − 1)dx
I e = 1
(2x − 3)2
dx I f = x√ 5 −
4x2 dx
I g =
x2(1 − 4x3) 13 dx I h =
x
2
3
2 − x53
−5dx I i =
(1 +
√ x)
1
4
√ x
dx
I j =
2 + 3x√
1 + 4x + 3x2dx I k =
arcsen(x)√
1 − x2 dx.
) I a =
2
3x
3
2
− 2√ x + C I b
= 1
25 (5x + 2)5
+ C
I c =
1
3√
(1 + x2
)3
+ C
I d = x5
5 − x
3
3 + C I e = −(2x −
3)
−1
2 + C
I f = −
√ 5 − 4x2
4 + C
I g = − 116
1 − 4x3 43 + C I h =
3
20
2 − x 53
−4+ C I i =
8
5
1 +
√ x 5
4 + C
I j =
√ 1 + 4x + 3x2 + C
I k =
arcsen2(x)
2 + C.
u = g(x)
f (u) = un f (u)
n + 1 undu =
un+1
n + 1 + C, n ̸= −1.
n ̸= −1
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f (u) = eu
e f (u) = eu
f ′(u) = eu f (u)
e
u
du = eu
+ C.
) I a =
ex√
ex − 5 dx I b =
xe−x2
dx
I c =
e
√ x
√ x
dx I d =
earctg(x)
1 + x2 dx
I b =
xe−x
2
dx
eudu = eu + C
I b
u = −x2.
du = −2xdx xdx
= −du
2 .
I b u
I n =
eu
−du
2
= −1
2
eudu i
= −12
eu + C.
u = −x2
x
u I n
I b = −12
e−x2
+ C.
senh(x)dx = cosh(x) + C
cosh(x)dx = senh(x) + C
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f (u) =
1
u
f (u) = ln(u)
f ′(u) = 1
u
1
u du = ln |u| + C, u > 0.
) I a =
1
4xdx I b =
e2x
e2x − 4dx I c =
dx
cos2(x)[3tg(x) + 1]
I a = 1
4x dx
1
u du = ln |u| + C, u > 0
i
I a = 1
4
1
xdx.
I a = 1
4 ln |x| + C
f (u) = au
f (u) = au
f ′(u) = au ln(a) audu =
au
ln(a) + C.
a) I a =
x222x3dx I b =
62x ln(6)dx I c =
(3x + 1)3(3x2+2x)dx
I a =
x222x
3
dx
audu =
au
ln(a) + C
I a
u = 2x3 du = 6x2dx.
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I n u
I n =
2u
du
6 =
1
6
2udu
= 1
6
2u
ln(2) + C.
I a
I a = 22x
3
6 ln(2) + C.
) I a =
ex (ex + 1)6 dx
I b =
dx
(1 + x2)arctg(x)
I c =
cos(2x)2sen(x) cos(x)
dx I d =
(3x + 1)1 − 6x − 9x2 dx
I e =
a
2
x
x2 dx, a > 0
I f =
3xexdx
I g =
e2x + a2x
2dx, a > 0.
) I a =
(ex + 1)7
7 + C I b = ln
|arctg(x)| + C
I c = 1
2 ln |sen(2x)| + C
I d = −1
6 ln |1 − 6x − 9x2| + C
I e = − a
2
x
2ln(a) + C
I f =
3xex
ln(3) + 1 + C
I g =
e4x
4 +
e2xa2x
ln(a) + 1 +
a4x
4ln(a) + C.
f (u) = sen(u)
f (u) = cos(u)
f ′(u) = −sen(u)
sen(u)du = −cos(u) + C.
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f (u) = cos(u)
f (u) = sen(u)
f ′(u) = cos(u)
cos(u)du = sen(u) + C.
a) I a =
sen(2x + 1)dx I b
=
sen(ln(x))
x dx
I c =
x2sen(2x3) +
sen(√
x)√ x − 2x
1 + x2
dx
I d =
ex cos(ex)dx
I e = x
√ x +
1
x
+ 1
2x
+ cos( 1
x)
x2 dx I f =
x cos(√
x2 + 1)
√ x2
+ 1
dx.
I a =
sen(2x + 1)dx
∫ sen(u)du = − cos(u) + C
u = 2x + 1
du = 2dx.
u
I n =
sen(u)
du
2 .
i
I n = 1
2 sen(u)du = −cos(u)
2 + C.
u = 2x + 1
x I a
I a = −cos(2x + 1)2
+ C.
I e =
x√
x +
1
x +
1
2x +
cos( 1x
)
x2
dx.
ii
I e =
x
1
2 dx +
1
xdx +
1
2xdx +
cos( 1x)
x2 dx.
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u =
1
x du = − 1
x2dx
I n = −
cos(u)du = −sen(u) + C.
x
I n = −sen
1
x
+ C.
I m = 2√
x3
3 + ln |x| − 2
−x
ln(2) + C.
I m + I n I e
I e = 2√ x3
3 + ln |x| − 2
−x
ln(2) − sen
1
x
+ C.
f (u) = tg(u)
f (u) = tg(u)
tg(u)du = ln | sec(u)| + C.
f (u) = tg(u)
tg(u)du = − ln | cos(u)| + C.
tg(u) = sen(u)
cos(u)
tg(u)du =
sen(u)
cos(u)du
v = cos(u) dv = −sen(u)du.
v
I n = −
dv
v = − ln |v| + C.
u
I n = − ln | cos(u)| + C.
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− ln | cos(u)| = ln |(cos(u))−1| = ln | sec(u)|.
tg(u)du = − ln |(cos(u))|
+ C = ln | sec(u)| + C.
I a =
tg(3x)dx I b =
tg(ln(x))
x dx
f (u) = cotg(u)
f (u) = cotg(u) cotg(u)du = −
ln |cosec(u)| + C.
f (u) = cotg(u)
cotg(u)du = ln |sen(u)| + C.
cotg(u) =
cos(u)
sen(u)
cotg(u)du =
cos(u)
sen(u)du
v = sen(u)
dv = cos(u)du
I n =
dv
v = ln |sen(v)| + C.
u
I n = ln |sen(u)| = − ln |cosec(u)|
+ C.
I a =
cotg(5x)dx I b =
cotg(e−x)
ex dx
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f (u) = sec(u)
f (u) = sec(u) sec(u)du = ln |
sec(u) + tg(u)| + C.
[sec(u) + tg(u)]
I n =
sec(u)
sec(u) + tg(u)
sec(u) + tg(u)
du =
sec2(u) + sec(u)tg(u)
sec(u) + tg(u)
du.
v = sec(u) + tg(u) dv =
[sec(u)tg(u) + sec2(u)] du
I n =
dv
v = ln |v| + C.
u
sec(u)du = ln | sec(u) + tg(u)|
+ C.
f (u) = cosec(u)
f (u) = cosec(u) cosec(u)du = ln
|cosec(u) − cotg(u)| + C.
cosec(u)du = − ln |cosec(u) + cotg(u)|
+ C.
f (u) = cosec(u)
[cosec(u) − cotg(u)]
I n =
cosec(u)
cosec(u) − cotg(u)cosec(u) − cotg(u)
du =
cosec2(u) − cosec(u)cotg(u)
cosec(u) − cotg(u)
du.
v = cosec(u) − cotg(u) dv =
[cosec2(u) − cosec(u)cotg(u)] du
I n =
dv
v = ln |v| + C.
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u
cosec(u)du = ln |cosec(u) − cotg(u)|
+ C.
I a =
sec
6
5x
dx I b =
x3
cos(x4)dx I c =
cosec
x3
dx.
f (u) = sec2(u)
f (u) = sec2(u) sec2(u)du = tg(u)
+ C.
d[tg(u)]
dx = sec2(u)
du
dx
f (u) = cosec2(u)
f (u) = cosec2(u)
cosec2(u)du = −cotg(u) + C.
d[cotg(u)]
dx = −cosec2(u) du
dx
I a =
e2x sec2
e2x
dx I b =
[1 − cosec(x)]2dx
f (u) = sec(u)tg(u)
f (u) = sec(u)tg(u) f (u)
sec(u)tg(u)du = sec(u) + C.
f (u)
d[sec(u)]
dx = sec(u)tg(u)
du
dx
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f (u) = cosec(u)cotg(u)
f (u) = cosec(u)cotg(u)
cosec(u)cotg(u)du = −cosec(u) + C.
d[cosec(u)]
dx = −cosec(u)cotg(u) du
dx
I a =
sen(2x)
cos2(2x)dx
I b =
excosec(ex)cotg(ex)dx
I a =
sen(x)cos(x)dx
I b =
cos2(x)dx
I c =
x cos(x2)dx
I d =
cotg(3x − 1)dx I e =
tg(πx)dx I f =
sen(x)
cos(x)dx
I g =
cosec(2x)cotg(2x)dx I h
=
sec2(x)√
1 + tg(x)dx I i =
dx
cos(2x)
I j =
tg
213x
sec
213x
dx
I k =
sec
2
(x)e
tg(x)
dx.
(∗)I a = −cos(2x)
4 + C I b =
1
2
sen(2x)
2 + x
+ C
I c =
sen(x2)
2 + C
I d =
1
3 ln |sen(3x − 1)| + C
I e = −1
π ln | cos(πx)| + C
I f = ln | sec(x)| + C
I g = −12
cosec(2x) + C I h =
2√
1 + tg(x) + C
I i =
1
2 ln | sec(2x) + tg(2x)| + C
I j = tg
3
1
3x
+ C
I k = etg(x) + C.
I a = sen
2(x)2
+ C
I a = −cos2(x)
2 + C
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I a =
ecotg(√ x)√
xsen2(√
x) +
3
x√
2 − ln(x2)
dx
I b =
2e
3√ x
3√
x2
e3√ x + 5
dx
I c =
x2ex
3
3 + 2ex3 +
2 + sen(3x)
cos2(3x)
dx
I d = −xsen(3x2 + 2)1 + cos(3x2 + 2)
+ x
2arctg(x3)
1 + x6
dx
I e =
ln(x)
x[3 + 2 ln2(x)]dx
I f =
2 + cotg[ln(x)]
xsen2[ln(x)] dx
I g =
cos{ln[sec(x)]} − cosec2(x)
cotg(x) dx
I h =
2 + tg1x
x2 cos
1x
dx
i
ii (x + 1)dx =
x2
2 + x + C.
u = x + 1
du = dx (x +
1)dx =
udu =
u2
2 + C =
(x + 1)2
2 + C.
I a = −2ecotg(√ x) − 3
√ 2 − ln(x2) + C
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I b = ln
e3√ x + 5
6+ C
I c =
1
6 ln
3 + 2ex3 + 23
tg(3x) + 1
3 sec(3x) + C
I d = 1
6
ln 1 + cos(3x2 + 2) + 16arctg(x3)2
+ C
I e = 1
4 ln |3 + 2 ln2(x)| + C
I f = −{2 + cotg[ln(x)]}2
2 + C
I g = sen{ln[sec(x)]} + ln |cotg(x)|
+ C
I h = −2 ln
sec
1
x
+ tg
1
x
− sec
1
x
+ C
